Strong Ellipticity Research Articles

A numeric-symbolic approach to the problem of localization of plastic flow

A method combining numerical and symbolic computations is presented for deciding whether or not a mechanical state given numerically in terms of its fourth-order tensor of moduli satisfies the localization condition, and more generally, any constitutive restriction on the tensor of moduli having a polynomial form. The idea underlying the method consists in posing the problem not as an optimization problem (minimizing the determinant of the acoustic tensor with respect to direction), but as a polynomial inequality. The polynomiality of the problem is then fully taken advantage of by using a powerful algorithm to solve polynomial inequalities: (a simplified version of) Collins' cylindrical algebraic decomposition algorithm. The method was implemented using the computer algebra system Mathematica, which provides accuracy-controlled symbolic and numerical computations and many built-in functions to handle polynomials. These features make it possible to determine the onset of localization more accurately and more reliably than with the usual optimization-based approaches, which are not guaranteed to converge to the global minimum. The potential of the method is illustrated by addressing loss of ellipticity and loss of strong ellipticity in the case of Gurson's porous material.

Boundary Integral Operators for Plate Bending in Domains with Corners

The paper studies boundary integral operators of the bi-Laplacian on piecewise smooth curves with corners and describes their mapping properties in the trace spaces of variational solutions of the biharmonic equation. We formulate a direct integral equation method for solving interior and exterior mixed boundary value problems on non-smooth plane domains, analyze the solvability of the corresponding systems of integral equations and prove their strong ellipticity.

Necessary and sufficient conditions for strong ellipticity of isotropic functions in any dimension

We consider hyperelastic stored energy functions in $\mathbb{R}^{n\times n}$ that areisotropic. We give necessary and sufficient conditions for the ellipticity of such functions.The present article is essentially a review of recent results on the subject.

A scalar boundary integrodifferential equation for eddy current problems using an impedance boundary condition

At low frequencies the time harmonic electromagnetic fields exterior to a lossy, highly conducting and possibly magnetic body can be described by the eddy current approximation of Maxwell’s equations with impedance or Leontovich [18] boundary conditions if the so called penetration depth is small. We show how to reduce this problem to a scalarr boundary integrodifferential equation on the surface Γ of the conductor. Strong ellipticity of the associated nonsymmetric, hypersingular operator is established. Convergence of O(h ) of the Ohmic losses for piecewise linear, continuous boundary elements is established theoretically and numerically.

Global Continuation in Displacement Problems of Nonlinear Elastostatics via the Leray-Schauder Degree

We obtain global solution continua of forced displacement problems of nonlinear elastostatics via a Leray-Schauder scheme. We adopt strong ellipticity as a bsic constitutive hypothesis. The usual Leray-Schauder approach, based in part upon the reduction of the boundary value problem to an operator equation for the zeros of a compact vector field, apparently fails. On the one hand, strong ellipticity alone is not enough to insure “invertibility/rdquo; of the principal, quasilinear part of the differential operator. More importantly, the physical requirement of local injectivity of the deformation and the associated growth of the stored-energy function dictate that ellipticity is not uniform. We demonstrate how to overcome these difficulties. Finally, with additional, physically reasonable restrictions on the stored-energy function, we obtain unbounded branches of classical, globally injective solutions.

Comments on loss of strong ellipticity in elastoplasticity

The loss of strong ellipticity is analyzed for a rate independent infinitesimal elastoplastic model. This local stability condition corresponds to the loss of positive definiteness of the symmetric part of the acoustic tensor. First, in the case of multisurface plasticity some expressions for the plastic hardening moduli are obtained for various bifurcation criteria. Next, explicit expressions for the critical plastic hardening modulus and the critical orientation are obtained in the case of single-surface plasticity (Hill type comparison solid). The analysis is based on a geometric method. Linear, isotropic elasticity, and a general nonassociative flow rule are assumed. However, the principal axes of the second order tensors of the plastic potential and yield surfaces gradients are coaxial. It is shown that, similar to the loss of ellipticity, the direction of the critical orientation is identical to one of the principal directions, except in the particular case where the gradient of the plastic potential and yield surfaces each have a double eigenvalue. In particular, explicit expression for the plastic hardening modulus, using the same geometric method, is also presented for the Raniecki type comparison solid. As an illustrative example, the critical orientation for the loss of strong ellipticity and the classical shear band localization (loss of ellipticity) are compared for axially-symmetric compression and tension.

Boundary integral method for thermoelastic screen scattering problem in ?3

We investigate a three-dimensional mathematical thermoelastic scattering problem from an open surface which will be referred to as a screen. Under the assumption of the local nite energy of the unied thermoelastic scattered eld, we give a weak model on the appropriate Sobolev spaces and derive equivalent integral equations of the rst kind for the jump of some trace operators on the open surface. Uniqueness and existence theorems are proved, the regularity and the singular behaviour of the solution near the edge are established with the help of the Wiener}Hopf method in the halfspace, the calculus of pseudodi!erential operators on the basis of the strong ellipticity property and Gas rding’s inequality. An improved Galerkin scheme is provided by simulating the singular behaviour of the exact solution at the edge of the screen. Copyright ( 2000 John Wiley & Sons, Ltd.

Optimal order collocation for the mixed boundary value problem on polygons

In usual boundary elements methods, the mixed Dirichlet-Neumann problem in a plane polygonal domain leads to difficulties because of the transition of spaces in which the problem is well posed. We build collocation methods based on a mixed single and double layer potential. This indirect method is constructed in such a way that strong ellipticity is obtained in high order spaces of Sobolev type. The boundary values of this potential define a bijective boundary operator if a modified capacity adapted to the problem is not 1. This condition is analogous to the one met in the use of the single layer potential, and is not a problem in practical computations. The collocation methods use smoothest splines and known singular functions generated by the corners. If splines of order 2m - 1 are used, we get quasi-optimal estimates in H m -norm The order of convergence is optimal in the sense that it is fixed by the approximation properties of the first missed singular function.

On strong ellipticity conditions for piezoelectric materials of the crystal classes 6 mm and 622

The conditions of strong ellipticity for piezoelectric materials of the crystal symmetry classes 6 mm and 622 are presented. These are derived using positivity conditions for quadratic, cubic and quartic polynomials. Moreover, a simplification of the slowness curve for the class 6 mm was carried out, enabling a simplified set of strong ellipticity conditions. Some piezomaterials were analysed demonstrating the feasibility of the proposed simplification.

Averaging theorems in finite deformation plasticity

The transition from micro- to macro-variables of a representative volume element (RVE) of a finitely deformed aggregate (e.g., a composite or a polycrystal) is explored. A number of exact fundamental results on averaging techniques, valid at finite deformations and rotations of any arbitrary heterogeneous continuum, are obtained. These results depend on the choice of suitable kinematical and dynamical variables. For finite deformations, the deformation gradient and its rate, and the nominal stress and its rate, are optimally suited for the averaging purposes. A set of exact identities is presented in terms of these variables. An exact method for homogenization of an ellipsoidal inclusion in an unbounded finitely deformed homogeneous solid is presented, generalizing Eshelby's method for application to finite deformation problems. In terms of the nominal stress rate and the rate of change of the deformation gradient, measured relative to any arbitrary state, a general phase-transformation problem is considered, and the concepts of eigenvelocity gradient and eigenstress rate are introduced. It is shown that the velocity gradient (and hence the nominal stress rate for rate-independent models) in an ellipsoidal region within an unbounded uniform and uniformly deformed solid, remains uniform when this region undergoes a uniform phase transformation corresponding to a constant eigenvelocity gradient. The generalized Eshelby tensor and its conjugate are defined and used to obtain the field quantities in an ellipsoidal inclusion which is embedded in an unbounded, uniformly deformed medium, leading to exact expressions for the concentration tensors and a set of identities relating these quantities. The Green functions for the rate quantities measured from a finitely deformed state, are formulated, and their properties are discussed. The details for calculating the Green function, Eshelby's tensor and its conjugate, and the concentration tensors, are presented. The relation between strong ellipticity and Green's function is examined in connection with the problem of the possible loss of stability of a uniformly stressed homogeneous solid. Exact values of the average nominal stress rate and the average velocity gradient, taken over an ellipsoidal region in a finitely deformed unbounded homogeneous solid, are obtained when arbitrary (variable) eigenvelocity gradients or eigenstress rates are prescribed in any region within the ellipsoid. It is shown that many results for single- and double-inclusion problems in linear elasticity, also apply to the finite-deformation rate problems, provided suitable kinematical and dynamical variables are used. The problem of the double inclusion is considered and exact expressions are given for the average field quantities, taken over the region between the two ellipsoidal domains, one containing the other, when arbitrary eigenvelocity gradients are prescribed within an arbitrary region contained in the inner ellipsoid. These results are further generalized to obtain exact expressions for the average field quantities taken over each annulus of a nested sequence of ellipsoidal regions within a finitely (uniformly) deformed infinite solid, where a constant distinct eigenvelocity gradient is prescribed on each annulus. These results are new, and should provide effective tools for homogenization of multi-component composites at finite deformations and rotations.

The Hölder and the Besov Regularity of the Density for the Solution of a Parabolic Stochastic Partial Differential Equation

In this paper we prove that the density p t ,x (y) of the solution of a white-noise-driven parabolic stochastic partial differential equation (SPDE) satisfying a strong ellipticity condition is 1 2 Lipschitz continuous with respect to (w.r.t.) t and 1 -e Lipschitz continuous w.r.t. x for all e ∈]0,1[ . In addition, we show that it belongs to the Besov space B 1 ,∞,∞ w.r.t. x . The proof is based on the Malliavin calculus of variations and on some refined estimates for the Green kernel associated with the SPDE.

Lack of strong ellipticity in Euclidean quantum gravity

Recent work in Euclidean quantum gravity has studied boundary conditions which are completely invariant under infinitesimal diffeomorphisms on metric perturbations. On using the de Donder gauge-averaging functional, this scheme leads to both normal and tangential derivatives in the boundary conditions. In the present paper, it is proved that the corresponding boundary value problem fails to be strongly elliptic. The result raises deep interpretative issues for Euclidean quantum gravity on manifolds with boundary.

Gauge Theories on Manifolds with Boundary

The boundary-value problem for Laplace-type operators acting on smooth sections of a vector bundle over a compact Riemannian manifold with generalized local boundary conditions including both normal and tangential derivatives is studied. The condition of strong ellipticity of this boundary-value problem is formulated. The resolvent kernel and the heat kernel in the leading approximation are explicitly constructed. As a result, the previous work in the literature on heat-kernel asymptotics is shown to be a particular case of a more general structure. For a bosonic gauge theory on a compact Riemannian manifold with smooth boundary, the problem of obtaining a gauge-field operator of Laplace type is studied, jointly with local and gauge-invariant boundary conditions, which should lead to a strongly elliptic boundary-value problem. The scheme is extended to fermionic gauge theories by means of local and gauge-invariant projectors. After deriving a general condition for the validity of strong ellipticity for gauge theories, it is proved that for Euclidean Yang–Mills theory and Rarita–Schwinger fields all the above conditions can be satisfied. For Euclidean quantum gravity, however, this property no longer holds, i.e. the corresponding boundary-value problem is not strongly elliptic. Some non-standard local formulae for the leading asymptotics of the heat-kernel diagonal are also obtained. It is shown that, due to the lack of strong ellipticity, the heat-kernel diagonal is non-integrable near the boundary.

On the Eigenvalues of the Fourth-Order Constitutive Tensor and Loss of Strong Ellipticity in Elastoplasticity

The eigenvalues of the fourth-order constitutive tangent modulus and the corresponding acoustic tensors are analyzed. Explicit expressions of the eigenvalues are made for the nonsymmetric tangent modulus tensor, and in the case of the deviatoric associative rule for the symmetric part of the tangent modulus and its acoustic tensor. In this context, a rate independent infinitesimal elastoplastic model is considered. The expressions of the plastic hardening modulus are summarized for the different local stability criteria (loss of second order work positiveness, loss of ellipticity, and loss of strong ellipticity). The critical hardening modulus and orientation are discussed in detail in the case of loss of ellipticity and loss of strong ellipticity. This analysis is based on the geometric method and linear, isotropic elasticity and deviatoric associative flow rule. In particular, the critical orientation for the loss of strong ellipticity and the classical shear band localization are compared.

On Hadamard stability for compressible viscoelastic constitutive equations

In this paper, the necessary and sufficient condition of the global Hadamard stabiltiy is formulated for isothermal and isotropic compressible viscoelastic constitutive equations. Two broad classes of quasilinear differential models (especially the Leonov class) and time-strain separable single integral models of either hyper- or nonhyper-viscoelastic type are considered. In order to derive an algebraic form of the stability condition, we modigy and then combine two mathematical procedures employed in formulating the Hadamard stability criteria for incompressible viscoelastic liquids and the strong ellipticity conditions for compressible elastic solids. We also briefly show that the concept of Hadamard stability is equivalent to the strong ellipticity of compressible elastic field equations. After restating inequalities for stability in terms of such separated variables as equivolumetric shear and volumetric components, we conclude that the stability criteria impose much more restrictive constraints on constitutive relations for compressible materials than for incompressible ones.

A reformulation of the strong ellipticity conditions for unconstrained hyperelastic media

The conditions for the strong ellipticity of the equilibrium equations of compressible, isotropic, nonlinearly elastic solids (established by Simpson and Spector [1]) are expressed in terms of the stored-energy function regarded as a function of the principal stretches. The applicability of this reformulation is illustrated with the help of two specific examples.

An engineering approach to generalized conjugate gradient methods and beyond

This is an elementary introductory paper (rather a tutorial) for generalized conjugate gradient (CG) methods. The emphasis is on “understanding” the approach of the now rather sophisticated CG-type methods and beyond. Practical aspects like normalization and smoothing are discussed. Then the development of different types of methods with their characteristic properties is presented. For a model of the Navier-Stokes equations that varies between strong and weak ellipticity, the methods are compared and their typical properties are demonstrated.

On the well-posedness of the initial value problem in elasto-plastodynamics for a linear comparison solid

This work deals with the well-posedness of the linearized initial value problem in solid elastoplastodynamics. By well-posedness we understand existence, uniqueness and continuous dependence of the solution with respect to initial and boundary data. The initial value problem is studied in its most general form using a technique from Kreiss [1]. The cases of elastic and elastic-plastic materials with associated and non-associated plastic flow rules are considered. The analysis is carried under the assumption that the material response remains within the same constitutive cone everywhere (i.e.; exclude loading/unloading subregions). A complete answer to the well-posedness question is given without the requirement of major symmetry of the constitutive tensor. First it is shown that in 2-D the necessary and sufficient condition for well-posedness is the strong ellipticity condition. Up to now this condition had been known only to be a sufficient condition.

On the approximation of singular integral equations by equations with smooth kernels

Singular integral equations with Cauchy kernel and piecewise-continuous matrix coefficients on open and closed smooth curves are replaced by integral equations with smooth kernels of the form(t−τ)[(t−τ)2−n2(t)e2]−1,e→0, wheren(t), t∈ Γ, is a continuous field of unit vectors non-tangential to Γ. we give necessary and sufficient conditions under which the approximating equations have unique solutions and these solutions converge to the solution of the original equation. For the scalar case and the spaceL2(Γ) these conditions coincide with the strong ellipticity of the given equation.

High-harmonic generation and correlated two-electron multiphoton ionization with elliptically polarized light.

The strong ellipticity dependence of correlated two-electron multiphoton ionization of neon and of the high-harmonic emission from argon (harmonic N=21) and neon (N=41) are reported. These measurements suggest a common underlying mechanism and are quantitatively consistent with a recently developed Keldysh-like model of high harmonic generation which treats the interaction between a newly freed electron and the ion core.